{
  "id": "1306.5100",
  "version": "v1",
  "published": "2013-06-21T11:23:30.000Z",
  "updated": "2013-06-21T11:23:30.000Z",
  "title": "Convergence and Quasi-Optimality of Adaptive FEM with Inhomogeneous Dirichlet Data",
  "authors": [
    "Michael Feischl",
    "Marcus Page",
    "Dirk Praetorius"
  ],
  "comment": "31 pages, 7 figures",
  "journal": "J. Comput. Appl. Math., 255 (2014), 481-501 (open access)",
  "doi": "10.1016/j.cam.2013.06.009",
  "categories": [
    "math.NA"
  ],
  "abstract": "We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with quasi-optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the L^2-projection or the Scott-Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott-Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-21T11:23:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "65N50"
    ],
    "keywords": [
      "inhomogeneous dirichlet data",
      "adaptive fem",
      "second order elliptic pde",
      "quasi-optimality",
      "edge-based residual error estimator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.5100F"
    }
  }
}